This parameter is used to reassign the desired trajectory \(q_d\)  according to the initial condition \(\varepsilon _{0} (0)\)  to the same hemisphere as the spacecraft quaternion so that the spacecraft attitude tracks the desired attitude by following the shortest path. Similar reassignment was used in [1]}, [2]}. Figure REF  illustrate these attitude errors in the Euclidean space \(\mathbb {R}^{4}\) . The illustration of the unit sphere \(\mathcal {S}^3\)  embedded in the space \(\mathbb {R}^4\)  is inspired by Fig. 3.10 in [3]}.
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